These discoveries must be attributed to the Platonic school in general ; for it is impossible to say With whom each originated. Some of advanced years frequented the school, as friends of its celebrated head, or out of respect for his doctrines; and others, chiefly young- persons, as disciples and pupils. Of the first class were Laodainus, Archytas, and Themtetus. Laodanius was one of the first to whom Plato communicated his method of analysis, be fore he made it public ; and he is said by Proclus to have profited greatly by this instrument of discovery. Archy tas was a Pythagorean of extensive knowledge in geome try and mechanics. Ile had a great friendship for Plato, and frequently visited him at Athens ; but in one of his voyages he perished by shipwreck. Thextetus was a rich citizen of Athens, and a Friend and fellow-student of Plato tinder Socrates, and Theodorus of Cyrcne, the geometer. He appears to have cultivated and extended the theory of the iegular solids.
The progress that geometry had then made, from the time of Hippocrates of Chios, required that the elements of the science should be new modelled. This was clone by Leon, a scholar of Neoclis, or Neoclide, a philosopher who had studied under Plato. To Leon has been ascribed also the invention of that part of the solution of a problem called its determination, which treats of the limits, or the cases in which it is possible. Eudoxus of Cnidus was one of the most celebrated of the friends and contemporaries of Plato. He generalized many theorems, and thereby greatly advanced geometry. It is believed that he culti vated the theory of the conic sections; and its invention has been attributed to him. He resolved the problem of the duplication of the cube ; and it is to be regretted that Ett tacit's, who despised his solution, has not thought fit to record it with the others, in his Commentary on Archi medes. Diogenes Laertius has attributed to him the in vention of curve lines in general ; from which we may in fer, that other curves than the conic sections were known in the school of Plato. Archimedes says, in the beginning of his treatise on the sphere and cylinder, that Eudoxus found the measure of the pyramid and cone, and that he bad treated of solids ; and others again have supposed, that he was the author of the theory of proportion as given in the fifth book of Euclid's Elements.
Passing over various geometers who are said to have distinguished themselves, but of whom hardly any thing more than the names are now known, we shall only men tion MerKechmus, and his brother Dinostratus. The for mer extended the theory of conic sections, insomuch that Eratosthenes seems to have given him the honour of their discovery, calling them the curves of illen.rchmus. His two solutions of the problem of two mean proportionals are a proof of his geometrical skill. Several discoveries have been given to Dinostratus ; but he is chiefly known by a property which he discovered of the Quadratri.r, a curve supposed to have been invented by Hippias of Elis.
After the death of Plato, his school was divided into two, which, upon some points, held opposite sentiments, but agreed in regarding a knowledge of the mathematics as absolutely necessary to such as would study philosophy. Thus the geometrical theories which had been culti vated with so much ardour in his life-time still continued to make progress. Xenocrates, the successor of Plato af
ter Speusippus, wrote on geometry and arithmetic. The principal geometers were all bred in the Platonic school, and among these probably we ought to reckon Aristxus, who is now little known, because his works are lost : we learn, however, from Pappus, that he was one of the an cients who hacl made the most progress in their sublime geometry. He composed a treatise on solid loci, in five books, and another on conic sections, also in five books, which last contained the greatest part of what was after wards given by Apollonius in the first four books of his work. Pappus placed this work after the conies of Apol lonius, in the order of study which he prescribed to his son : This spews that it was a profound theory, and sup nosed the doctrine of conics to be previously known.
He is reputed to have been the friend and preceptor of Eu clid.
The progress of geometry among the Peripatetics was not so brilliant as it had been in the school of Plato, but the science was by no means neglected. The successor of Aristotle composed several works relating to mathematics, and particularly a complete history of these sciences down to his own time : there were four books on the history of geometry, six on that of astronomy, and one on that of arith metic. What a treasure this would have been, had we now possessed it ! The next remarkable epoch in the history of geometry, after the time of Plato, was the establishment of the school of Alexandria, by Ptolemy Lagus, about SOO years before the Christian xra. This event was highly propitious to learning in general, and particularly to every branch of mathematics then known; for the whole was then culti vated with as much attention as had been bestowed on geo metry alone in the Platonic school. It was here that the celebrated geometer Euclid flourished, under the first of the Ptolemies: his native place is not certainly known, but lie appears to have studied at Athens, under the disciples of Plato, before he settled at Alexandria. Pappus, in the introduction to the seventh book of his Collections, gives him an excellent character, describing him as gentle, mo dest, and benign towards all, and more especially such as cultivated and improved the mathematics. There is an anecdote recorded of Euclid, which seems to shew he was not much of a courtier: Ptolemy Philadelphns having asked him whether there was any easier way of studying geometry than that commonly taught; his reply was, " there is no royal road to geometry." This celebrated man composed treatises on various branches of the ancient mathematics, but he is best known by his Elements, a work on geometry and arithmetic, in thirteen books, under which he has collected all the elementary truths of geome try which had been found before his time. The selection and arrangement have been made with such judgment, that, after a period of 2000 years, and notwithstanding the great additions made to mathematical science, it is still generally allowed to be the best elementary wcrk on geo metry extant. Numberless treatises have been written since the revival of learning, some with a to improve, and others to supplant the work of the Greek geometer : but in this country, at least, they have been generally ne glected and forgotten, and Euclid maintains his place in our schools.